Binomial Theorem - Formula, Expansion, Properties, Proof, Examples

In the simplest terms, the binomial theorem is a shortcut or formula that tells you how to expand mathematical expressions into a sum of terms. The most basic example of an expression in the binomial theorem is (a+b)^n 

• The ‘n' letter is a positive integer, like 1, 2, 3,... 
• a and b are real numbers, like 3, 2.5,..., or even variables, such as x, y. 

Now, what does this solve, you ask?

With it, you won’t multiply the [removed]a+b) by itself n times over and over again. That’s too much work. The Binomial Theorem gives you a ready-made formula that tells you

• How many terms will there be
• What each term looks like 
• What number goes in front of them 

Here is a numerical explanation of the binomial theorem. Let’s say, we want to find (2 + 1)^3, which we already know is 3^3 = 27. Here 27 is the final answer. The Binomial Theorem shows how we got there - like your cricket replay!

You can think of it this way. You saw the scoreboard show 27 runs, but you missed the action. Were they all the boundaries? Was it a six? 

Now, the Binomial Theorem is your match highlight reel. It breaks the total down ball by ball. In this case, term by term.

See the breakdown here of the binomial expansion. 

(2+1)^3 = 2^3 + 3 * 2^2 * 1 + 3 * 2 * 1^2 + 1^3 = 8 + 12 + 6 + 1 = 27

So, here every term is a different play. 

• 8 from the first term. That could have been a powerful boundary shot.
• 12 from the second term. Maybe, two sixes back-to-back.
• 6 from the third term. Three well-placed twos.
• 1 from the final term. Like a quick single to finish the over

Just like the cricket scorecards, you will see that the Binomial Theorem breaks down the (a+b)ⁿ into different terms. This Binomial expansion acts as a cricket scorecard. It breaks down (a+b)ⁿ into individual terms. This theorem works for any variable, and not just numbers.

Importance of the Binomial Theorem

You will build a competitive foundation through understanding the binomial theorem. 

1. The JEE Main syllabus includes the Binomial theorem for a positive integral index, as well as general and middle terms. You may be asked about finding the 5th term or the coefficient of a power. 
2. For your exams, it also carries around 6 to 7 marks. So definitely do not miss this. 
3. If you master the Binomial Theorem, you could secure good marks, as the questions for this chapter are pretty predictable. 

In Chapter 7 of your NCERT textbook, you will find an introductory description of the binomial theorem instead of a formal definition. It starts by explaining how you, as a student, have “learnt how to find the squares and cubes of binomials like a + b and a – b. Using them, we could evaluate the numerical values of numbers like (98)2 = (100 – 2)2, (999)3 = (1000 – 1)3, etc. However, for higher powers like (98)5, (101)6, etc., the calculations become difficult by using repeated multiplication. This difficulty was overcome by a theorem known as binomial theorem. It gives an easier way to expand (a + b) n, where n is an integer or a rational number.”

Some important terms that support the Binomial Theorem 

1. Binomial coefficient - These numbers can tell you how many of each term you will have. They're represented as ⁿCᵣ (sometimes called "n choose r")
2. Pascal's Triangle - It’s a helpful triangle of numbers that gives you the binomial coefficients.

        Row 0:            1

        Row 1:          1   1

        Row 2:        1   2   1

        Row 3:      1   3   3   1

        Row 4:    1   4   6   4   1

        Row 5:  1   5  10  10   5   1

Here, every number is a combination.

Row is n, while position r gives you ( nr )Binomial Expansion - The long expression you get after using the binomial theorem.

If n is a positive integer, and a,b are real numbers, the binomial theorem for one specific term states:

(a+b)n=Σ nCra(n−r)b′ (from r=0 to n)

Each term in the expansion has a binomial coefficient nCr. The calculation is nCr=n!/(r!(n−r)!)

These coefficients have interesting properties. 

1. nCr= nC(n−r) (Symmetry)
2. nC0= nCn=1 *(Edge Case)
3. nC1= nC(n−1)=n  (First Items)

To break it down further, let’s understand the terms more. 

Symmetry : This means you’re choosing two items from five, which provides the same result as leaving three items from five.

Example: ⁵C₂ = ⁵C₃ = 10

Edge cases: Here, there's only one way to choose nothing. And also, one way to select everything.

Example: ⁿC₀ = ⁿCₙ = 1

First items: This means there are exactly n ways to choose 1 item from n items.

Example: ⁿC₁ = ⁿC(ₙ₋₁) = n

1. Sum of Coefficients:
   (a+b)n if (a=1,b=1)=(1+1)n=2n
2. Alternating Sum of Coefficients:
   (a+b)n at a=1,b=−1(1−1)n=0
3. Middle Terms:

• If n is even: one middle term at r=n2

If n is odd: two middle terms at r=(n−1)2 and (n+1)2

The binomial expansion of (a+b)n has exactly (n+1) terms. Each term looks like

T(r+1)= nCra(n−r)br

But, what do all these actually mean?

1. (n + 1) terms? When you expand (a+b)n, you always get one more term than the power.

Example:

• (a+b)2 → 3 terms
  
• (a+b)3→ 4 terms

2. What does a term look like? Each term follows this pattern. 

• T(r+1) means Term number (r + 1). We start counting r from 0.
• nr is the coefficient in front
• a(n−r) is the power of a that is going down

br is the power of b and it goes up

Here is a basic example of the binomial theorem. 

(x+y)3=x3+3x2y+3xy2+y3

Here is how you can quickly apply this formula for the binomial theorem. 

By expanding (x+y)3, we will get four terms. The powers of x start high and eventually go down. The powers of y start low but keep going up. 

The numbers in front (1, 3, 3, 1) come from Pascal’s Triangle or the combination formula ( 3r) 

So instead of multiplying (x+y)(x+y)(x+y)(x + y)(x + y)(x + y)(x+y)(x+y)(x+y) the long way, we use this shortcut to get the same answer quickly and neatly.

So, instead of multiplying (x+y)three times, we use this shortcut to get the same answer quickly.